Posted by ayan on Wednesday, April 17, 2013 at 12:09am.
x ^ 2 + x y + x = 81
x ( x + y + 1 ) = 81 Divide both sides by x
x + y + 1 = 81 / x
y ^ 2 + x y + y = 51
y ( y + x + 1 ) = 51
y ( x + y + 1 ) = 51 Divide both sides by y
x + y + 1 = 51 / y
x + y + 1 = x + y + 1
81 / x = 51 / y Multiply both sides by x y
81 x y / x = 51 x y / y
81 y = 51 x Divide both sides by 81
81 y / 81 = 51 x / 81
y = 51 x / 81
y = 3 * 17 * x / ( 3 * 27 )
y = 17 x / 27
Now put this value in formula :
x ( x + y + 1 ) = 81
x ( x + 17 x / 27 + 1 ) = 81
x ( 27 x / 27 + 17 x / 27 + 1 ) = 81
x ( 44 x / 27 + 1 ) = 81
44 x ^ 2 / 27 + x = 81
44 x ^ 2 / 27 + x - 81 = 0 Multiply both sides by 27
44 x ^ 2 + 27 x - 2187 = 0
The exact solutions of this equation are :
x = - 81 / 11 and x = 27 / 4
For x = - 81 / 11
y = 17 x / 27 = - 51 / 11
x + y = - 81 / 11 - 51 / 11 = - 132 / 11 = - 12
For x = 27 / 4
y = 17 x / 27 = 17 / 4
x + y = 27 / 4 - 17 / 4 = 44 / 4 = 11
x + y is positive so :
x = 27 / 4 , y = 17 / 4
x + y = 11
For x = 27 / 4
y = 17 x / 27 = 17 / 4
x + y = 27 / 4 + 17 / 4 = 44 / 4 = 11